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Matrix functions which are not scalar coefficient power series expansions.
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Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather matrix coefficients:
$$f(bf A) = sum_k=0^infty bf C_k A^k, bf C_k,Ain M^ntimes n$$
I suppose that it would be more difficult to show that $bf A$ diagonalizable $Rightarrow f(bf A)$ diagonalizable (in the general case) as it obviously is in the special case $forall k: bf C_k=c_kbf I$ (scalar coefficients). Since identity commutes with all matrices.
(Is it even true? (secondly))
matrices soft-question power-series diagonalization
asked Jul 31 at 15:45
mathreadler
13.5k71857
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up vote
2
down vote
favorite
Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather matrix coefficients:
$$f(bf A) = sum_k=0^infty bf C_k A^k, bf C_k,Ain M^ntimes n$$
I suppose that it would be more difficult to show that $bf A$ diagonalizable $Rightarrow f(bf A)$ diagonalizable (in the general case) as it obviously is in the special case $forall k: bf C_k=c_kbf I$ (scalar coefficients). Since identity commutes with all matrices.
(Is it even true? (secondly))
matrices soft-question power-series diagonalization
asked Jul 31 at 15:45
mathreadler
13.5k71857
I kind of doubt that $A$ diagonalizable implies $f(A)$ diagonalizable unless there were additional constraints on the $C_k$
– Dark Malthorp
Jul 31 at 19:07
@DarkMalthorp Yes I also very much doubt it, but I am also not sure if many such functions make sense. Most matrix functions I have seen are power series expansions of normal one variable real valued functions which always have scalar coefficients.
– mathreadler
Jul 31 at 19:16
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather matrix coefficients:
$$f(bf A) = sum_k=0^infty bf C_k A^k, bf C_k,Ain M^ntimes n$$
I suppose that it would be more difficult to show that $bf A$ diagonalizable $Rightarrow f(bf A)$ diagonalizable (in the general case) as it obviously is in the special case $forall k: bf C_k=c_kbf I$ (scalar coefficients). Since identity commutes with all matrices.
(Is it even true? (secondly))
matrices soft-question power-series diagonalization
asked Jul 31 at 15:45
mathreadler
13.5k71857
Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather matrix coefficients:
$$f(bf A) = sum_k=0^infty bf C_k A^k, bf C_k,Ain M^ntimes n$$
I suppose that it would be more difficult to show that $bf A$ diagonalizable $Rightarrow f(bf A)$ diagonalizable (in the general case) as it obviously is in the special case $forall k: bf C_k=c_kbf I$ (scalar coefficients). Since identity commutes with all matrices.
(Is it even true? (secondly))
matrices soft-question power-series diagonalization
asked Jul 31 at 15:45
mathreadler
13.5k71857
edited Jul 31 at 15:52
asked Jul 31 at 15:45
mathreadler
13.5k71857
asked Jul 31 at 15:45
mathreadler
13.5k71857
asked Jul 31 at 15:45
mathreadler
13.5k71857
13.5k71857
I kind of doubt that $A$ diagonalizable implies $f(A)$ diagonalizable unless there were additional constraints on the $C_k$
– Dark Malthorp
Jul 31 at 19:07
@DarkMalthorp Yes I also very much doubt it, but I am also not sure if many such functions make sense. Most matrix functions I have seen are power series expansions of normal one variable real valued functions which always have scalar coefficients.
– mathreadler
Jul 31 at 19:16
add a comment |Â
I kind of doubt that $A$ diagonalizable implies $f(A)$ diagonalizable unless there were additional constraints on the $C_k$
– Dark Malthorp
Jul 31 at 19:07
@DarkMalthorp Yes I also very much doubt it, but I am also not sure if many such functions make sense. Most matrix functions I have seen are power series expansions of normal one variable real valued functions which always have scalar coefficients.
– mathreadler
Jul 31 at 19:16
I kind of doubt that $A$ diagonalizable implies $f(A)$ diagonalizable unless there were additional constraints on the $C_k$
– Dark Malthorp
Jul 31 at 19:07
I kind of doubt that $A$ diagonalizable implies $f(A)$ diagonalizable unless there were additional constraints on the $C_k$
– Dark Malthorp
Jul 31 at 19:07
@DarkMalthorp Yes I also very much doubt it, but I am also not sure if many such functions make sense. Most matrix functions I have seen are power series expansions of normal one variable real valued functions which always have scalar coefficients.
– mathreadler
Jul 31 at 19:16
@DarkMalthorp Yes I also very much doubt it, but I am also not sure if many such functions make sense. Most matrix functions I have seen are power series expansions of normal one variable real valued functions which always have scalar coefficients.
– mathreadler
Jul 31 at 19:16
add a comment |Â
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I kind of doubt that $A$ diagonalizable implies $f(A)$ diagonalizable unless there were additional constraints on the $C_k$
– Dark Malthorp
Jul 31 at 19:07
@DarkMalthorp Yes I also very much doubt it, but I am also not sure if many such functions make sense. Most matrix functions I have seen are power series expansions of normal one variable real valued functions which always have scalar coefficients.
– mathreadler
Jul 31 at 19:16